# American Institute of Mathematical Sciences

November  2020, 3(4): 279-300. doi: 10.3934/mfc.2020018

## Support vector machine classifiers by non-Euclidean margins

 School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China

* Corresponding author: Qi Ye

Received  January 2020 Revised  May 2020 Published  June 2020

Fund Project: The first author is supported by the Innovation Project of Graduate School of South China Normal University. The second author is supported by the Natural Science Foundation of Guangdong Province (2019A1515011995, 2020B1515310013) and the National Natural Science Foundation of China (11931003)

In this article, the classical support vector machine (SVM) classifiers are generalized by the non-Euclidean margins. We first extend the linear models of the SVM classifiers by the non-Euclidean margins including the theorems and algorithms of the SVM classifiers by the hard margins and the soft margins. Specially, the SVM classifiers by the $\infty$-norm margins can be solved by the 1-norm optimization with sparsity. Next, we show that the non-linear models of the SVM classifiers by the $q$-norm margins can be equivalently transferred to the SVM in the $p$-norm reproducing kernel Banach spaces given by the hinge loss, where $1/p+1/q = 1$. Finally, we illustrate the numerical examples of artificial data and real data to compare the different algorithms of the SVM classifiers by the $\infty$-norm margin.

Citation: Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018
##### References:
 [1] B. E. Boser, I. M. Guyon and V. N. Vapnik, A training algorithm for optimal margin classifiers, in Proceedings of the Fifth Annual Workshop on Computational learning theory, ACM, (1992), 144–152. doi: 10.1145/130385.130401.  Google Scholar [2] P. Bühlmann and S. van de Geer, Statistics for High-Dimensional Data. Methods, Theory and Applications, Springer Series in Statistics, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-20192-9.  Google Scholar [3] L. Chen and H. Zhang, Statistical margin error bounds for l1-norm support vector machines, Neurocomputing, 339 (2019), 210-216.  doi: 10.1016/j.neucom.2019.02.015.  Google Scholar [4] C. Cortes and V. Vapnik, Support-vector networks, Machine Learning, 20 (1995), 273-297.  doi: 10.1007/BF00994018.  Google Scholar [5] R. Der and D. Lee, Large-margin classification in banach spaces, Journal of Machine Learning Research - Proceedings Track, 2 (2007), 91-98.   Google Scholar [6] I. Ekeland and T. Turnbull, Infinite-Dimensional Optimization and Convexity, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1983.   Google Scholar [7] T. Hastie, R. Tibshirani and M. Wainwright, Statistical Learning with Sparsity: The Lasso and Generalizations, Monographs on Statistics and Applied Probability, 143. CRC Press, Boca Raton, FL, 2015.   Google Scholar [8] L. Huang, C. Liu, L. Tan and Q. Ye, Generalized representer theorems in Banach spaces, Anal. Appl. (Singap.), (2019). doi: 10.1142/S0219530519410100.  Google Scholar [9] O. L. Mangasarian, Arbitrary-norm separating plane, Operations Research Letters, 24 (1999), 15-23.  doi: 10.1016/S0167-6377(98)00049-2.  Google Scholar [10] J. Platt, Sequential minimal optimization: A fast algorithm for training support vector machines., Google Scholar [11] L. Q. Qi, H. B. Chen and Y. N.Chen, Tensor Eigenvalues and Their Applications, Advances in Mechanics and Mathematics, 39. Springer, Singapore, 2018. doi: 10.1007/978-981-10-8058-6.  Google Scholar [12] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.   Google Scholar [13] B. Schölkopf and A. J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, The MIT Press, Cambridge, 2001.   Google Scholar [14] G. H. Song and H. Z. Zhang, Reproducing kernel Banach spaces with the $\ell^1$ norm Ⅱ: Error analysis for regularized least square regression, Neural Comput., 23 (2011), 2713-2729.  doi: 10.1162/NECO_a_00178.  Google Scholar [15] G. H. Song, H. Z. Zhang and F. J. Hickernell, Reproducing kernel Banach spaces with the $\ell^1$ norm, Appl. Comput. Harmon. Anal., 34 (2013), 96-116.  doi: 10.1016/j.acha.2012.03.009.  Google Scholar [16] I. Steinwart and A. Christmann, Support Vector Machines, Information Science and Statistics, Springer, New York, 2008.  Google Scholar [17] V. N. Vapnik, The Nature of Statistical Learning Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2440-0.  Google Scholar [18] Y. S. Xu and Q. Ye, Generalized mercer kernels and reproducing kernel banach spaces, Mem. Amer. Math. Soc., 258 (2019). doi: 10.1090/memo/1243.  Google Scholar [19] H. Yang, X. Yang, F. Zhang, Q. Ye and X. Fan, Infinite norm large margin classifier, International Journal of Machine Learning and Cybernetics, 10 (2019), 2449-2457.  doi: 10.1007/s13042-018-0881-y.  Google Scholar [20] H. Z. Zhang, Y. S. Xu and J. Zhang, Reproducing kernel Banach spaces for machine learning, J. Mach. Learn. Res., 10 (2009), 2741-2775.  doi: 10.1109/IJCNN.2009.5179093.  Google Scholar [21] L. Zhang and W. Zhou, On the sparseness of 1-norm support vector machines, Neural Networks, 23 (2010), 373-385.  doi: 10.1016/j.neunet.2009.11.012.  Google Scholar [22] J. Zhu, S. Rosset, R. Tibshirani and T. J. Hastie, 1-norm support vector machines, in Advances in Neural Information Processing Systems, (2004), 49–56. Google Scholar

show all references

##### References:
 [1] B. E. Boser, I. M. Guyon and V. N. Vapnik, A training algorithm for optimal margin classifiers, in Proceedings of the Fifth Annual Workshop on Computational learning theory, ACM, (1992), 144–152. doi: 10.1145/130385.130401.  Google Scholar [2] P. Bühlmann and S. van de Geer, Statistics for High-Dimensional Data. Methods, Theory and Applications, Springer Series in Statistics, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-20192-9.  Google Scholar [3] L. Chen and H. Zhang, Statistical margin error bounds for l1-norm support vector machines, Neurocomputing, 339 (2019), 210-216.  doi: 10.1016/j.neucom.2019.02.015.  Google Scholar [4] C. Cortes and V. Vapnik, Support-vector networks, Machine Learning, 20 (1995), 273-297.  doi: 10.1007/BF00994018.  Google Scholar [5] R. Der and D. Lee, Large-margin classification in banach spaces, Journal of Machine Learning Research - Proceedings Track, 2 (2007), 91-98.   Google Scholar [6] I. Ekeland and T. Turnbull, Infinite-Dimensional Optimization and Convexity, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1983.   Google Scholar [7] T. Hastie, R. Tibshirani and M. Wainwright, Statistical Learning with Sparsity: The Lasso and Generalizations, Monographs on Statistics and Applied Probability, 143. CRC Press, Boca Raton, FL, 2015.   Google Scholar [8] L. Huang, C. Liu, L. Tan and Q. Ye, Generalized representer theorems in Banach spaces, Anal. Appl. (Singap.), (2019). doi: 10.1142/S0219530519410100.  Google Scholar [9] O. L. Mangasarian, Arbitrary-norm separating plane, Operations Research Letters, 24 (1999), 15-23.  doi: 10.1016/S0167-6377(98)00049-2.  Google Scholar [10] J. Platt, Sequential minimal optimization: A fast algorithm for training support vector machines., Google Scholar [11] L. Q. Qi, H. B. Chen and Y. N.Chen, Tensor Eigenvalues and Their Applications, Advances in Mechanics and Mathematics, 39. Springer, Singapore, 2018. doi: 10.1007/978-981-10-8058-6.  Google Scholar [12] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.   Google Scholar [13] B. Schölkopf and A. J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, The MIT Press, Cambridge, 2001.   Google Scholar [14] G. H. Song and H. Z. Zhang, Reproducing kernel Banach spaces with the $\ell^1$ norm Ⅱ: Error analysis for regularized least square regression, Neural Comput., 23 (2011), 2713-2729.  doi: 10.1162/NECO_a_00178.  Google Scholar [15] G. H. Song, H. Z. Zhang and F. J. Hickernell, Reproducing kernel Banach spaces with the $\ell^1$ norm, Appl. Comput. Harmon. Anal., 34 (2013), 96-116.  doi: 10.1016/j.acha.2012.03.009.  Google Scholar [16] I. Steinwart and A. Christmann, Support Vector Machines, Information Science and Statistics, Springer, New York, 2008.  Google Scholar [17] V. N. Vapnik, The Nature of Statistical Learning Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2440-0.  Google Scholar [18] Y. S. Xu and Q. Ye, Generalized mercer kernels and reproducing kernel banach spaces, Mem. Amer. Math. Soc., 258 (2019). doi: 10.1090/memo/1243.  Google Scholar [19] H. Yang, X. Yang, F. Zhang, Q. Ye and X. Fan, Infinite norm large margin classifier, International Journal of Machine Learning and Cybernetics, 10 (2019), 2449-2457.  doi: 10.1007/s13042-018-0881-y.  Google Scholar [20] H. Z. Zhang, Y. S. Xu and J. Zhang, Reproducing kernel Banach spaces for machine learning, J. Mach. Learn. Res., 10 (2009), 2741-2775.  doi: 10.1109/IJCNN.2009.5179093.  Google Scholar [21] L. Zhang and W. Zhou, On the sparseness of 1-norm support vector machines, Neural Networks, 23 (2010), 373-385.  doi: 10.1016/j.neunet.2009.11.012.  Google Scholar [22] J. Zhu, S. Rosset, R. Tibshirani and T. J. Hastie, 1-norm support vector machines, in Advances in Neural Information Processing Systems, (2004), 49–56. Google Scholar
Examples of Euclidean margins and non-Euclidean margins are illustrated. The black line is the decision boundary. The distance of the gap between the two red dashed lines is the margin. We can see the difference between these two classifiers
The difference between hard margin and soft margin solutions
The geometrical interpretation of distance from a point to a plane
Maximal margin classifiers by 2 norm and $\infty$ norm in the case of 2 distinct points. We can see the difference between the abilities to obtain sparsity
The geometric explanation for infinite solutions for the SVM classifier by $\infty$-norm margin
The geometric explanation for unique solution for the SVM classifier by $\infty$-norm margin
The geometrical interpretation of the kernel tricks. In the original space, the data set is not linear separable, see Figure 6a. After the feature mapping using kernel function, the data set is linear separable in higher dimensional space, see Figure 6b. Finally, the decision boundary in higher dimensional space can be projected into the original space and becomes the non-linear boundary in the original space
Noiseless and noisy data sets. The classes overlaps in noisy data set
Three classifiers obtained by three different optimizations when $C = 0.5$, $\lambda = 2$
Two classifiers obtained by two different optimizations when $C = 0.1$, $\lambda = 10$
The comparison between handwritten digits 6 and 9. The first row is 6 and the second row is 9
The comparison between handwritten alphabets O and Q. The first row is O and the second row is Q
The results of experiments on MNIST. The row Linear SVM classifier by ∞-norm margin is the result obtained by (9) where $\lambda = 0.1$. The row Kernel SVM classifier by ∞-norm margin is the result obtained by (15) where $\lambda = 0.1$. The kernel function used here is Gaussian kernel $K(\boldsymbol x, \boldsymbol x') = \exp \frac{-\|\boldsymbol x- \boldsymbol x'\|_{2}^{2}}{\sigma^2}$ where $\sigma = 9.6$
 MNIST Training Errors Test Errors Sparsity Linear SVM classifier by ∞-norm margin 0/9939 5/1967 654/785 Kernel SVM classifier by ∞-norm margin 0/9939 4/1967 1760/9939
 MNIST Training Errors Test Errors Sparsity Linear SVM classifier by ∞-norm margin 0/9939 5/1967 654/785 Kernel SVM classifier by ∞-norm margin 0/9939 4/1967 1760/9939
The results of experiments on Handwritten Alphabets Database. The row Linear SVM classifier by ∞-norm margin is the result obtained by (9) where $\lambda = 0.001$. The row Kernel SVM classifier by ∞-norm margin is the result obtained by (15) where $\lambda = 0.001$. The kernel function used here is Gaussian kernel $K(\boldsymbol x, \boldsymbol x') = \exp \frac{-\|\boldsymbol x- \boldsymbol x'\|_{2}^{2}}{\sigma^2}$ where $\sigma = 2600$
 Handwritten Alphabets Training Errors Test Errors Sparsity Linear SVM> classifier by ∞-norm margin 0/7000 69/3000 401/785 Kernel SVM classifier by ∞-norm margin 0/7000 14/3000 1339/7000
 Handwritten Alphabets Training Errors Test Errors Sparsity Linear SVM> classifier by ∞-norm margin 0/7000 69/3000 401/785 Kernel SVM classifier by ∞-norm margin 0/7000 14/3000 1339/7000
 [1] Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048 [2] Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021005 [3] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [4] Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010 [5] Min Ji, Xinna Ye, Fangyao Qian, T.C.E. Cheng, Yiwei Jiang. Parallel-machine scheduling in shared manufacturing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020174 [6] Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176 [7] Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443 [8] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [9] Dmitry Dolgopyat. The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces. Journal of Modern Dynamics, 2020, 16: 351-371. doi: 10.3934/jmd.2020014 [10] Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2021, 17 (1) : 29-50. doi: 10.3934/jimo.2019097 [11] Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117 [12] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [13] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [14] Kengo Nakai, Yoshitaka Saiki. Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1079-1092. doi: 10.3934/dcdss.2020352 [15] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [16] Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 [17] Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230 [18] Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361 [19] Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145 [20] Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002

Impact Factor:

## Tools

Article outline

Figures and Tables